Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, then we know that $$d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$$
for each compact subset $K\subset U$, also that $$d_{2}(K,\partial U)=d_{2}(\hat{K}_U,\partial U)$$
for each compact subset $K\subset U$, this equality also holds for all norm on $\mathbb{C}^n$?
Any help would be appreciated.
$\hat{K}_U= \{z \in U: |f(z)| \leq \sup_K |f|, \forall f\in \mathcal{O}(\Omega)\}$: holomorphically convex hull of $K$.
$d_{\infty}(A,B)=\mathrm{inf}\{\Arrowvert a-b\Arrowvert_{\infty}:a\in A,b\in B\}$
$d_{2}(A,B)=\mathrm{inf}\{\Arrowvert a-b\Arrowvert_{2}:a\in A,b\in B\}$
$\Arrowvert z\Arrowvert_{2}=\left\{\sum_{j=1}^{n}|z_j|^2\right\}^{1/2}$ , $\Arrowvert z\Arrowvert_{\infty}=\max\{|z_1|,\ldots,|z_n|\}$: particular norms.